Apparatus for playing a game

ABSTRACT

Apparatus for playing a game including a pack of cards in which: each card has a numerical face value which is a whole number; the cards are divided into a plurality of groups that each card in any group has the same digit root as the other cards in the same group; and the cards in each group carry common identifying means signifying that the cards belong to that group.

[0001] THIS INVENTION relates to improvements in the construction and organisation of playing cards and board games which are used for recreational and educational purposes. In particular, the invention to be described here is about an innovative method of organising numerical playing cards based on the concept of the digit root of the face value of the numbers inscribed on the cards.

[0002] Definition & Notation: In simple terms, the digit root¹ of a number is the result of successively adding up the digits of the number until one arrives at a single number which is less than 10. For example, the number 56 has the digit root of 2, since 5+6=11 and 1+1=2. Similarly, the digit root of the number 34219 is 1 since 3+4+2+1+9=19 and 1+9=10, 1+0=1.

[0003] In algebraic terms, the digit root of a non-negative integer, N is k, if there exists a non-negative integer, r such that N=9r+k, where k is an integer and 0≦k≦9.

[0004] To facilitate the expression of the notion or concept of digit roots as described above and as will be used throughout the description of this invention, the notation

[0005] ‘

’

[0006] is hereby introduced to symbolise ‘has the digit root . . . ’.²

[0007] Thus to say that the number 56 has a digit root of 2, we simply write:

[0008] 56

2. Similarly, we write 34219

1 and so on.

[0009] Following the non-technical definition of the digit root of a number given above, it is not difficult to notice that the digit root of any whole number lies between 0 and 9 inclusive. In other words, the digit root of any number in the decimal base is either 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. In fact, the only number whose digit root is 0 (zero) is 0 itself. That means that every other whole number has a digit root which is either 1, 2, 3, 4, 5, 6, 7, 8 or 9.

[0010] Any set of cards organised according to the digit root of the face value of the number on each of the cards is what is referred to here as Ichongiri's Numerical Playing Cards. The expression Digit Root Numerical Cards or simply Digit Root Cards could also be accepted to mean and therefore substitute for Ichongiri's Numerical Playing Cards, especially where there is no room for ambiguous interpretation.

[0011] Organisation and Description of the Cards: The numerical cards for which this invention is a guide are constructed and organised by calculating the digit root of any whole number greater than 0, where a whole number is the same as the face value of the numerical card. Numerical cards whose face values yield the same digit root are grouped together and treated as a class. Thus, the first one hundred and eight numbers (in the decimal base) organised in terms of their digit root classes are as follows: TABLE 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

[0012] It should be pointed out from Table 1 above that all the decimal numbers along a given column each has the same digit root.

[0013] Ichongiri's Numerical Cards can also be organised accordingly as a sub-set of the content of Table 1. For example, a set of numerical cards organised in terms of their digit roots where each of the numbers involved is:

[0014] a) even;

[0015] b) odd;

[0016] c) prime;

[0017] d) square numbers; or

[0018] e) square-free numbers etc.

[0019] Table 2 below contains the first 54 odd integers arranged according to the digit root method: TABLE 2 1 2 3 4 5 6 7 8 9 1 11 3 13 5 15 7 17 9 19 29 21 31 23 33 25 35 27 37 47 39 49 41 51 43 53 45 55 65 57 67 59 69 61 71 63

[0020] Table 3 below contains the first 54 even integers arranged according to the digit root method: TABLE 3 1 2 3 4 5 6 7 8 9 10 2 12 4 14 6 16 8 18 28 20 30 22 32 24 34 26 36 46 38 48 40 50 42 52 44 54 64 56 66 58 68 60 70 62 72

[0021] A whole section in this specification will be devoted to numerical cards whose face values are all prime numbers and which are arranged in terms of their digit roots as well as a set of games devised by the inventor to illustrate their use.

[0022] Further Description of the Numerical Cards:

[0023] Each numerical card which this invention describes may be a polygonal-shaped tangible or virtual medium with a flat surface made of/from paper, plastic, liquid crystal or any similar material capable of being used to construct a playing card on which is inscribed:

[0024] i) a whole number greater than 0 (zero), called the face value of the card expressed in words or in figures in any language; and

[0025] ii) where membership of the face value of a card is solely constructed or determined on the basis of the digit root of the face value of the card encoded/imprinted in some distinguishable form and/or capable of being so interpreted;

[0026] The inscription/encoding as described in (i) and (ii) above could be done using pencil, pen or any form of stylus used for writing. It could also be done using a printer, a computer or any other convenient device suitable to a particular environment.

[0027] More explicitly and for purposes of uniformity it is recommended that the card be rectangular shaped and that every numerical card should have inscribed on it:

[0028] a) four (4) digit root sign symbols (

), one at each corner of the rectangular shaped card.

[0029] b) the digit root of the face value of the card which identifies it as a member of a digit root class expressed in figures. For example, the card which bears the face value 78 has a digit root of 6 and identifies it a member of all the non-negative integers whose digit root is 6. On this card, therefore, the digit root sign with the associated digit root will be displayed as ‘

6’ on the four corners of the number card;

[0030] c) a rectangular shaped region within the larger rectangular-shaped numerical card. This inner region or section of the card should bear the colour or differentiating symbol or code assigned to the digit root class to which the numerical card belongs; where the numerical card happens to occupy a special position within the digit root class, for example, if the face value of the numerical card is the same as its digit root, any marker used to index this special position in the form of symbol, notation or picture must be outlined in the colour of the digit root class or should have it as its background;

[0031] d) three or two letters of the English alphabet unique to the digit root class to which the card belongs.

[0032] The following table presents each digit root class together with its assigned colour or colour code. TABLE 4 Digit Root Class Assigned Colour 1 Blue 2 Brown 3 Green 4 Pink 5 Purple 6 Black 7 Red 8 Orange 9 Metallic Ash/Grey

[0033] These are some of the features which every Ichongiri numerical card may satisfy in terms of inscription.

[0034] Types of Ichongiri's Numerical Cards

[0035] A set or group of Ichongiri's Numerical Cards will be described as mixed if in addition to the cards being organised in terms of the digit roots of their face values, the face values themselves are either all positive integers (as given in table 1) or a combination of types of integers which are not mutually exclusive. For example, a set of Ichongiri's Numerical Cards in which all the numbers are odd (as given in table 3) contains both odds which are composites and those which are prime and therefore will constitute a mixed set or pack.

[0036] On the other hand, a set or group of Ichongiri's Numerical Cards will be described as specific if in addition to the cards being organised in terms of the digit roots of their face values, the face values themselves belong to a mutually exclusive category. For example, a set or pack of Ichongiri's numerical cards in which the face value of every number is prime satisfies the criteria of being specific.

[0037] Physical Specification of the Cards:

[0038] An Ichongiri card could be made from ordinary paper, carton, cardboard or a plastic or marble. Where the play environment is computer-mediated, it could be made using a liquid crystal which will be used as a medium for making the necessary inscriptions.

[0039] The exact material to be used the possible materials listed above and the exact dimensions of the cards should be chosen in line with the budget, aim and unique circumstances of the target audience or users.

[0040] But for purposes of illustration and by way of recommendation, it is suggested that an Ichongiri numerical card should be made from

[0041] a Satin 300 gsm Cardboard (paper);

[0042] Each card should measure about 88 mm by 57 mm;

[0043] The front side should be Litho print with varnish;

[0044] The reverse side should be Litho print with gloss laminate.

[0045] The edges of the cards should be properly trimmed and prepared in a way that will not pose any health hazards to anyone using the cards; these must adhere in general with the health and safety regulations (if any) governing the production of playing cards in any country or state where the cards are to be produced and or used.

[0046] Even though the cards could be used without any box or container, it is recommended that each pack be put into a convenient box or container

[0047] For purposes of illustration and by way of recommendation, it is suggested that for a pack of between 54 and 56 cards specified as above, the box or container should be made from a 400 micron folding board which should measure about 60 mm by 20 mm by 90 mm. The outside should be painted in green with UV varnish.

[0048] Section A: The Mixed Type

[0049] To exemplify a mixed pack or group, the set of all positive integers between 1 and 54 will be used in this specification.

[0050] For the sub-pack consisting of numbers 1-54, the cards whose face values are the first 9 integers (1 to 9) are assigned representational roles. Each of these cards in addition to the obligatory and non-obligatory features earlier outlined, should have a picture of drawing of two similar human heads joined neck to neck and symmetrical about a given line. Any other distinguishing symbol or code should suffice.

[0051] FIGS. 1-9 contain a possible copy pictures of the cards arranged according to digit root sameness. Formal drawings of the cards or sample thereof will be submitted later with necessary alterations and changes.

[0052] For a pack or sub-pack consisting of numbers 55-108, the cards whose face values are the first 9 integers (55 to 64) should be assigned representational roles. Each of these cards in addition to the obligatory and non-obligatory features earlier outlined, should have a picture of drawing of two similar human heads joined neck to neck and symmetrical about a given line. Any other distinguishing symbol or code should suffice.

[0053] As a rule, the number of cards which should assume a representational role within a pack or sub-pack must be a multiple of the number of distinct digit root classes which are used to organise the cards. (An exception to this general rule is the case of a pack in which all the face values are prime numbers. More on this later in Section B).

[0054] Representational cards are conferred with special powers and privileges and could be used as wild cards to suit the particularities of certain games.

[0055] As a rule, it is recommended that within the mixed type, each standard pack or sub-pack should contain 54 numerical cards: 6 from each of the 9 digit root classes.

[0056] However, for educational purposes, play and or teaching could be carried out using numerical cards so arranged with any convenient pack or combination of packs or even sub-groups/digit root classes within a pack or combination of packs. For the practical task of making or organising the cards, it is recommended that the number of cards in a pack³ should be a multiple of 9 to guarantee equal distribution of cards among the digit root classes. In practice, two or more standard packs of 54 could be combined to form an extended pack. A pack will be described as ‘extended’ if the number of numerical cards in the pack is greater than 54 whether.

[0057] Uses of the Numerical Cards (Mixed and Specific)

[0058] The present invention: Ichongiri's Numerical Cards are conceived principally as a tool for teaching and supporting the learning of mathematics and mathematically-based concepts within various contexts and especially within a teaching-learning environment. Whilst most of the games which could be played using the cards are in the main educational and pedagogic, the cards could be used for games which are recreational in nature and therefore, arguably, non-educational.

[0059] From the pedagogic and educational point of view, Ichongiri's numerical cards and games could be used to teach, support and facilitate the teaching of the following concepts and activities within a teaching-learning environment, for example a school:

[0060] Geometrical Topics: Drawing of triangles, Pythagoras Theorem, Plotting of points on a grid, a geometrical approach to finding rational irrational numbers. Drawing graphs of the form x±y=k, where k is a non-negative integer less than 9; Graphical solution of simultaneous linear equations.

[0061] Topics in Arithmetic: A practical and intuitive approach to prime number recognition and determination of nontrivial composite integers; Addition, Subtraction and Multiplication modulo 9 & Modulo 18, General Digit Root Arithmetic; Buying and Selling, The notion of Bargaining & Trade-offs, Banking and Revenue. Divisibility and Discrimination of Composites using Digit Root Arithmetic.

[0062] Statistics & Probability: A practical introduction to combinatorics (combinations and permutations), tallying and frequency distribution, probability of events, equally likely outcomes and finding the range.

[0063] School Algebra: Construction and solution of specific Diophantine Equations, Linear Graphs, Introduction to Hyperbolic Curves of the form XY=K, (where k is a constant), Change of Subject Formula, Symmetric Property of variables and Equivalence Relations.

[0064] Group Theory: Properties of a Group (Closure, Associativity, Inverse, Identity Element), Multiplicative Inverses, Commutative property, Order of a group, Cyclic groups and Isomorphism.

[0065] Recreational Mathematics: Casting of 9's, Number games, ‘Magimatics’—Numerical ‘Divination’ or Mind Reading, Competition and Strategic Thinking. Application of the method of casting of 9's in Accounting and Cataloguing.

[0066] Culture and Civics: Family Tree, Hierarchy within the Family, Patriarchy, Generational Continuity, Respect and Responsibility within the traditional African society and (arguably) within any traditional society in the world.

[0067] Tools and Objects embodied in the Invention: The tools and objects which could be used and/or manipulated to make use of the invention referred to in this specification are the face values on the numerical cards, the digit roots, the two- or three-letters on card (where applicable), any pictures or photos (where applicable) as well as the symbol or colour on the card. In most cases, however, it is the face values together with the associated digit roots (and symbol or colour code) that will be used playing devised games.

EXAMPLES

[0068] For the board game of Uhie na Aka (to be described later), the feature of a card made use of is its digit root which serves as a co-ordinate for either the horizontal or vertical axes. In addition, there is the board as well as the chips to be described later which are used.

[0069] For games used to teach or support school mathematics (in particular number sense and confidence), it is the face values which are the principal tools or objects used. In some of them, it is a combination of the face value and the digit root or colour of the card;

[0070] For games used to introduce and support the arithmetic and algebra of digit root, only the digit roots are used;

[0071] For games involving the drawing of points (for example, Strategic Acquisition to be described later), digit roots of cards are used together with graph paper, pencil or pen (which could be procured locally).

[0072] The letters on a card, are a mnemonic device for memorising the nominal positions of the English alphabet (and in deed).

[0073] Next we present a selection of games playable using Ichongiri's Numerical Cards. Except where it is otherwise stated, it is assumed that the games to be presented are based on a Mixed pack of Ichongiri's Numerical Cards which consists of the first 54 positive integers organised in terms of their digit root classes. Table 5 below contains these numerical cards. TABLE 5 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

[0074] Illustrations:

[0075] First is a series of games in which, mutatis mutandis, the player who earns the highest number of cards and/or the cards whose face values give the highest total is the winner. Example:

[0076] 1. Name of the game: Your Best 2

[0077] Number of Players: 2, 3, 4, 5 or 6

[0078] Aim of the Game: To earn as many cards as possible from the pool.

[0079] Materials needed: A well shuffled pack of Ichongiri's Numerical Cards (Mixed) consisting of the first 54 positive integers.

[0080] How it is Played:

[0081] 1) A pack of Ichongiri's numerical cards is placed face down on the active platform. Starting with the player sitting to the right of the player who shuffled the cards, each player selects 4 cards from the general pool in turn.

[0082] 2) After each player has collected the initial 4 cards, the first person who picked up, selects 2 out of the 4 cards in his or hands and places them on the active platform in a way which reveals the face value and digit root of each of the cards. This is repeated by each player in accordance with the order in which they picked up the initial 4 cards.

[0083] 3) The player whose 2 turned over cards yield the greatest total or sum collects all the turned over cards. The earned cards are to be banked—that is, put aside and not to be mixed with the cards in the hands of the player who has earned them.

[0084] 4) Each player then collects 2 more cards from the general pool (in any order). It is then the turn of the second player to pick up the 4 initial cards to be the first person to reveal any 2 of the cards in his or her hands, then every other player will do same in the order in which they had picked up the initial cards (with the first player to pick up the initial cards being the last to reveal his/hers at this stage).

[0085] 5) Then steps 3) and 4) above are repeated until there are no more cards to be picked up equally by the players. At this stage the game ends.

[0086] 6) The winner of the set is determined by either:

[0087] a) counting the number of cards which each player earned and had banked with the player having the highest number of cards being declared the winner; or

[0088] b) by adding the face value of the cards earned by each player and declaring the player with the highest face value total the winner;

[0089] Rules Guiding the Game:

[0090] i) The 2 numerical cards to be turned over must not have the same digit root;

[0091] ii) A player is permitted to turn over any two cards of his/her choice, whether or not they are the highest;

[0092] iii) A player who fails to turn over any 2 cards which are acceptable (probably because they could not make the allowed combinations) cannot be allowed to earn any cards no matter the sum of the two turned over cards;

[0093] iv) In the case where the number of players is two, and in the event of (iii) above, the other player is allowed to collect all the turned over cards provided he/she makes the correct combination (that is, different digit roots) even if his/her total is smaller than the other player who failed to produce the correct combinations;

[0094] v) Should both of the players or all of the players become unable to make the acceptable combinations (in terms of the digit root criterion), then the digit root criterion will be waived for that phase of play. As a result, only the player(s) with the greatest total should collect or earn all the turned over cards.

[0095] vi) Depending on the age and ability as well as the aim of the game, the use of a calculator may be allowed.

[0096] vii) In case of two players and both having the same total with the correct combinations, each person earns his/her own contribution.

[0097] Variations:

[0098] Several variations of the above game could be played. Examples of these variations are:

[0099] Variant 1) where the turned over cards must be one even and the other odd (with all the rules intact);

[0100] Variant 2) where the number of initial cards picked up by a player is 6; three of which must be turned over in turn and where one of the turned over cards must be even, the other an odd composite and the third one a prime number;

[0101] It is clear that using the tools and objects embodied in this invention and applying the general functions of principles guiding the use of privileged Ichongiri Numerical Cards given on page, various other games could be constructed which the limitations of space do not permit their elaboration in this specification.

[0102] Next we consider Ichongiri's Numerical Cards of a Specific Nature. For this the prime numbers have been selected.

[0103] Section B: Specific Type (Prime)

[0104] Based on the definition of the digit root of a number earlier given, it could be shown very easily that apart from 3, that any number whose digit root is 3 is divisible by 3 and therefore cannot be prime.

[0105] Similarly, if the digit root of a number is 6, then the number is divisible by either 2 or 3 or both and therefore cannot be prime and if the digit root of a number is 9 that number is divisible by 9 and therefore is not prime. 2 is the only even prime and 3 is the only prime integer whose digit root is 3.

[0106] Consequently, all prime integers greater than 3 can only fall into any of the six digit root classes namely: 1, 2, 4, 5, 7 and 8. Table 6 below presents all prime integers greater than 3 but less than 100 in terms of their digit root class membership. TABLE 6 1 2 4 5 7 8 19 11 13 5 7 17 37 29 31 23 43 53 73 47 67 41 61 71 83 59 79 89 97

[0107] Notation and Further Description of the Cards:

[0108] Table 7 presents all the first 56 prime integers in terms of their digit root class membership. TABLE 7 1 2 3 4 5 7 8 19 2 3 13 5 7 17 37 11 31 23 43 53 73 29 67 41 61 71 109 47 103 59 79 89 127 83 139 113 97 107 163 101 157 131 151 179 181 137 193 149 223 197 191 173 211 167 241 233 271 191 229 239 277 251 227

[0109] Designation:

[0110] Because 2 is the only prime integer which is even and 3 the only prime whose digit root is 3, 2 and 3 will for purposes of this invention be referred to as the Strangers. (More on the strangers later).

[0111] Removing them from table 7 we have table 8: TABLE 8 1 2 4 5 7 8 19 11 13 5 7 17 37 29 31 23 43 53 73 47 67 41 61 71 109 83 103 59 79 89 127 101 139 113 97 107 163 137 157 131 151 179 181 173 193 149 223 197 191 191 211 167 241 233 271 227 229 239 277 251

[0112] The prime pack of INC which this specification refers to will be made up of the content of table 8 plus the Strangers:—2 and 3. The associated colour code for each of the digit root classes is the same as earlier assigned in this specification. That is, for the digit root class 1, the colour Blue has been used as a distinguishing marker. For digit root class 2 we have Brown, for digit root class 4 Pink, for digit root class 5 we have Purple, for digit root class 7 Red and for digit root class 8 we have Orange. FIGS. 4, 5 and 6 contain all the numbers involved at level 1 of the primes. The two strangers in addition to having their digits inscribed on them, must be differentiated from the others in terms of colour and in terms of the picture or symbol on them. The map of Africa has been chosen as the symbol for 3 while the map of the world (the globe) is the symbol for 2.

[0113] From table 8 above, it is clear that every digit root class is made up of 9 cards each of which is unique and distinct. For each digit root class, four Ichongiri cards will be designated as super cards called Ichie, Nwata, Nne and Nna which mean Ancestor, Child/Offspring, Mother and Father respectively in Igbo⁴ language

[0114] In numerical terms, Ichie is the smallest of the prime numbers (in terms of face value) of cards within a digit root class. It represents the first, primordial and ancestral link of the numbers within a digit root class. In fact, it is easy to see that every prime number within a digit root class is 18 times a number plus the Ichie of that digit root class. Thus, there is a sense in which every prime member of a digit root class carries within itself ancestral gene—the Ichie gene, as it were.

[0115] On their part, the Nwata, Nne and Nna super cards in this set of Numerical Cards represents the 7^(th), 8^(th) and 9^(th) prime members (in terms of face value) of their digit root class. Whilst the Ichie remains the first of the members of a digit root (or colour) class, the position of Nwata, Nne and Nna super cards may be altered when bigger and larger numbers are used in playing the games suggested here.

[0116] The meanings and place of Ichie, Nwata, Nne and Nna within the Igbo traditional (and arguably, the African) society are very significant. Some of these and the power relations which underpin the mechanics of their operation could be gleaned from some of the games. For example, the notion of the child/offspring (mainly the male child) within the Igbo society is the future of the family and the assurance for its continuity, essentially, is the rationale behind the redemptive demands of the game of Nwaka.

[0117] The Strangers: Apart from the 54 Ichongiri just described, the prime pack of cards under consideration consists of two extra-ordinary cards, namely, 2 and 3. As already mentioned, none of these two cards falls into our system of classification. But they are prime numbers—indeed, ‘special’ primes. Therefore, for purposes of completeness, they are added and described as the two Strange cards or simply the Strangers. They are part of our picture cards. 3 is called Obodo which stands for Nation while 2 bears the name Uwa which means the World. The games in which they are involved attest to their superior and fortune-changing powers. When these two Strange cards are added to the Standard pack of 54 Ichongiri cards, we have a Full or Complete pack of 56 cards.

[0118] In strict hierarchical terms, the most powerful card is the Uwa (the World). This is that which was before any of us came into being. It is the totality of physical and non-physical systems and relationships within which we operate. It is represented by the map of the world (the globe).

[0119] The next most powerful Card is Obodo (nation, country or continent). This is represented by the map of Africa of Africa. It is part of and exists within the Uwa, it is bigger than any individual or group of individuals and therefore has an over-riding, veto-like influence should the need arise. Next is Ichie, followed by Nna, Nne and then Nwata⁵.

[0120] An Important Note: As much as possible, all the names, symbols and letters on any Ichongiri Numerical Card should be preserved by any authorised user of the cards. However, where a company, person or group of persons licensed/and or authorised to use any of the rights which may accrue from the invention of these cards, may be permitted to translate part or all of the names (herein expressed in Igbo language) into English or a more convenient language. For this to happen, written permission must be obtained from the inventor or any person(s) authorised to act on his behalf.

[0121] Unless otherwise stated to the contrary, all further references to Ichongiri's Numerical Cards refer to the Prime Pack just described.

[0122] Uses of the Card Games:

[0123] The Numerical Cards under consideration could be put into good use in a number of contexts or scenarios. These contexts could be divided into the family, the schools, colleges/universities and the cultural community or society.

[0124] The Family: Parents and guardians will find the Ichongiri Numerical Cards and Games a helpful aid in teaching their children and young ones ideas, concepts, practices and rituals which relate to the family and how it is generally structured. Even though the names of the super cards are expressed in Igbo language and that the construction of the general themes African driven, people and families from non-African societies will find the concepts very relevant and the cards an effective pedagogic tool. It is difficult to imagine or conceive of any culture or society in which the concepts of the Ichie, Nna, Nne and Nwata as herein defined are absent.

[0125] Schools: Teachers and parents at the primary and secondary school levels of education will find the cards and some of the games very useful. The games could be used to teach basic concepts and themes in the areas of numeracy and basic skills, social studies/civics, government etc. For example, the game called A little to the right, a little to the left could be used to give practice in the area of mental addition and subtraction. It could also be customised to suit the age level and ability of the pupils or students

[0126] Colleges and Universities: The cards and some of the games will also be suitable for college and university level students of mathematics and other quantitative courses. The main scheme of which these cards and games are a part of its implementation was developed using the concept of Group Theory. A careful look at the digit root classes of {1, 2, 4, 5, 7, and 8} will show that the set forms a commutative group of order 6 under the digit root operation as defined here. Furthermore, some of the suggested games make use of the notions of multiplicative inverses and the identity element. More importantly, students at this level can construct their own set of Ichongiri numerical cards (with 9 digits, say) to familiarise themselves more with prime numbers with a view to recognising them.

[0127] Cultural Communities and Societies: The cards and the games are also an invaluable tool in the hands of elders and custodians of traditional culture, especially the African traditional cultures and other cultures whose mode of social and economic organisation are essentially patriarchal. In particular, this set of cards together with the some of the suggested games is a must for any teacher and student of Igbo language and culture.

[0128] Illustration of Some Games:

[0129] 1. Currency Earner

[0130] Number of players: 2, 3, 4, 5 or 6.

[0131] Aim of the game: Using initial cards or talents or capital to earn as much currency as possible. At the end of a game, the winner is the player who has won or earned the greatest amount of currency say money using the initial cards, capital or talent.

[0132] For this game, a standard pack of 54 cards is used. (That means that the two strangers:—Uwa and Obodo are not used). After shuffling the cards, the dealer deals a minimum of 6 cards to each of the players one at a time. The rest of the pack is placed face down and forms the general pool. The initial card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0133] In this game, a card is playable if it

[0134] a) belongs to the same digit root class (that is, if it is of the same colour as the topmost card or the turned over card);

[0135] b) is a super card,

[0136] c) is a strange card.

[0137] Rules:

[0138] When a player can't play, he/she must lose or surrender one of his cards in his/her hands. This is placed underneath the current card in exchange for a pick of the next card on top of the general pool.

[0139] Any cards earned in the course of a game must be banked and kept separate from the cards or talents in the hands of the player who has earned them.

[0140] At the time of stopping a game, each player's set of earned cards in a bank is added together (manually or electronically) the total score ascertained, and the proper designation based on prior agreement declared. Whoever has the highest total is the winner of that particular game.

[0141] Exceptions:

[0142] No player is allowed to earn using a Nwata⁶ super card;

[0143] Only an Ichie could be used to earn an Ichie.

[0144] For young children who may not be able to add large numbers which may be involved or who may be unable to use the calculator correctly, the winner could be ascertained by the person/player who earns the greatest number of individual cards. Counting could be facilitated here by pairing the cards.

[0145] It may be necessary to decide on how many games should make up a set. In this case, a player's amount earned in any particular game is recorded and the total ascertained from time to time. For example, in an n-game set, each player's total earnings per game are added together and recorded. Whoever is the first to earn W (where W is a pre-agreed value ) over the agreed n games becomes the overall set winner. For example, W=5,000 (units of money (say the Pound Sterling) or talents) in which case the first to earn £5,000 units over a number of games becomes the winner. Non-earned cards do not count.

[0146] Scoring: The default rule in scoring is that the worth of a card is its face value when it comes to counting, unless otherwise stated in which case agreed⁷ values or units of currency could be agreed upon before the commencement of the game.

[0147] End of game: The game comes to an end when the last card of the general pool is collected or turned over or when any one of the players exhausts all the initial capital in his/her possession. At this point, every player surrenders all non-earned cards in his/her hands.

[0148] Variant: More to Earn:

[0149] This is a variant of the Currency Earner described above. But for the More to Earn, a playable card must be more than the face value of the turned over card. It should also be mentioned, that here Nwata is allowed to earn another card so long as it is greater than the turned over card in face value. The exception to this rule is that Nwata is still not allowed to earn an Ichie (even though the Nwata's face value is always greater than that of the Ichie).

[0150] Comment: This game and its variants can be played by and with children as young as four and could be used to introduce the concept of buying and selling, savings, the principle of enjoying the benefits of ones labour. The More to Earn variant could be used to introduce children to the mathematical notions of greater than and ordering of integers. Furthermore, the restriction placed on Nwata in the Currency Earner could serve as starting point in teaching division of labour and role expectations within a community.

[0151] 2. Family Assembly:

[0152] Number of players: 2, 3, 4, 5 or 6. (Each given at least 6 Ichongiri cards)

[0153] Aim of the game: To be the first to assemble the full family members of a digit root family (or colour class). Essentially, we are talking about a player being able to assemble (or have in his or her hands), an Ichie, Nwata, Nne and Nna of the same colour or digit root class.

[0154] Alternatively, any three of the four Super cards within a digit root (colour) class plus the Uwa or Obodo strange cards will suffice. However, two of the four super cards within a digit root class plus the two strangers will not be accepted as a full lineage.

[0155] Rules:

[0156] Any other card in possession of the player at the time of stopping the game and which does not form part of the Full lineage will count against the player. Thus, it is possible for a player who has a Full Lineage to delay announcing same to the group if he/she has higher numbers or other strange cards which do not currently form a Full family set. This will depend, of course, on what the other players are holding.

[0157] Possibilities:

[0158] a) Outright win by the announcer;

[0159] b) Win by another player (as a result of the odd members attached to the Full Set);

[0160] c) A draw.

[0161] However, it is permissible for a player to ‘buy’ what another player has got rid of which the first player feels he/she needs in order to assemble a full lineage. This system of ‘buying’ must be conducted under well defined guidelines.

[0162] The Guidelines:

[0163] First, a player can only ‘buy’ or ‘exchange’ what he/she needs, if it is his/her turn to play and the turned over card is still the one he/she needs;

[0164] Second, as a price for getting what he/she wants, he/she must pick the topmost card from the general pool.

[0165] Third, he/she will not be permitted to release or get rid of a card at this point. Thus, the art of collecting what he/she needs and then picking a card from the general pool counts as a turn. He/she must wait for their next turn to be able to play or shout Full lineage as the case may be.

[0166] How it is played: A card is playable if it matches the turned over card by:

[0167] (i) by colour; (no matter the difference between the turned over card and the card in hand);

[0168] (ii) by number (if the face value of the card in hand is greater than the face value of the turned over card).

[0169] (iii) by a Special privilege (that is, where any of the Super cards (Ichie, Nne, Nna, Nwata) or any of the Strangers (Uwa, Obodo) could be placed on top the preceding card not minding whether or not rules (i) and (ii) are met.

[0170] A player who can't play must go to the general pool to pick a card.

[0171] End of game: The game is be brought to an end immediately a player satisfies the requirement of producing the 4 super cards in any one digit root class and shouts Full Lineage!.

[0172] Scoring: Any cards which do not form a full lineage must count against the player in whose hands they are found. The default rule for scoring also applies here.

[0173] Comments: Notions and themes like family membership⁸, family tree and lineage, negotiation, compromise and co-operation could be taught or introduced using this game. More generally, the relationship between the individual members of society, the nation and the world could also be explored and the implications analysed using this game.

[0174] 3. A Little to the Right, a Little to the Left!

[0175] Number of players: 2, 3 or 4, 5 or 6

[0176] Aim of the game: To be the first player to release all the cards in ones hands.

[0177] How it is Played:

[0178] A full or complete pack of 56 cards is used. The dealer deals 6 cards to each of the players one at a time. The rest of the pack is placed face down and forms the general pool. The topmost card called the initial card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0179] In this game, a card is playable if its face value lies within some specified number above or below the face value of the turned over card, the digit root class (or colour) notwithstanding. For example, players may decide that a card in hand is playable if it is within plus or minus 20 of the turned over card. Using this illustration, if the turned over card is 73, then the player whose turn it is to play may play any card which lies between (73−20 and 73+20), that is any prime number within the pack which lies between 53 and 93 inclusive. Note that the range could be much higher or much lower depending on the age, ability and wish of the students or pupils with whom the game is being used as a facilitating tool for learning and teaching.

[0180] A player who does not have a playable card or who chooses not to play picks a card from the general pool.

[0181] Rules:

[0182] At the time of stopping a game, the number of cards and its face value in one's hands will count against the person. In the case of more than two players, the less the value in your hands, the better is your ranking or position.

[0183] The Ichie and the Nwata can always be used to decree a range which one wants and can always be placed on top of any turned over card.

[0184] End of game: The game comes to an end when one of the players surrenders or releases all the cards in his or her hands.

[0185] Scoring: The first to release all his/her cards is the winner. However, where three or more players are involved, there may be the need to rank the players and marks assigned accordingly. For example, the winner (that is the first to release all his/her cards) could be assigned 100 points, the person with the next smallest total (of the face value of the cards in his her hands when the game ends) will then be assigned 80 points, next 60 points etc.

[0186] 4. Nwaka⁹

[0187] Aim of the game: To have in your hands as many Nwata as possible, while every other card is got rid off.

[0188] Number of players: 2, 3, 4, 5 or 6

[0189] How it is Played:

[0190] A standard pack of 54 cards is used. The dealer deals a minimum of 6 cards to each of the players one at a time. The rest of the pack is placed face down and forms the general pool. The topmost card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0191] In this game, a card is playable if it

[0192] a) belongs to the same digit root class (that is, is of the same colour as the topmost card or the turned over card);

[0193] b) is a super card.

[0194] Whenever the topmost card is Nwata, the one whose turn it is to play is under obligation to redeem the Nwata. Nwata is redeemed by the player exchanging the Nwata with either an Nne or an Nna That means that only an Nne or an Nne can redeem a Nwata. The redeeming Nne or Nna does not necessarily have to be from the same family colour (or digit root class) as the child who is being redeemed.

[0195] A player whose turn it is to redeem the Nwata but who for any reason fails to do so must go to the pool to pick a card (as a form of penalty). The chance to redeem the Nwata is then passed on to the next player (to the right of the previous player who had failed to redeem Nwata) and so on.

[0196] However, if after 3 cycles or rounds of failing to redeem the Nwata, normal play continues. That is, the next player can any card whose digit root is the same as that of the Nwata about to be lost. In this case, no one picks or takes the Nwata.

[0197] Winning: The first person to get rid of all other cards but a Nwata or a number of Nwatas declares: “Game, Continuity Assured”.

[0198] Scoring:

[0199] In the case of 3 or more players, whatever any player has in his/her hands which, at the end of play, is not a Nwata counts against him or her. The numerical or face value of the cards remaining in each player's hands (except a Nwata) is added together. The person whose total is highest is the first to go out or the one to earn the least point.

[0200] An example of a scoring system which could be used in the case of 3 or more players is as follows: Points earned Rank 100  1^(st) (The winner) 60 2^(nd) 40 3^(rd) 30 4^(th) 20 5^(th) 10 6^(th)

[0201] End of the game: This game ends following the allocation of points as earlier agreed before the beginning of play. Where the scoring system is the same as or similar to the one suggested in the above table, more than a game is likely to be played. Where such is the case the overall winner will be the player who is the first to earn 300 points or more.

[0202] Variants: Variants of this game could be created by a conscious decision by players to privilege any of the other super cards or even an ordinary card (iti) in the way that has been done to Nwata here and names assigned accordingly.

[0203] 5. The Goldbach¹⁰ Target

[0204] Aim of game: To be the first to collect and declare two distinct Ichongiri cards (primes) each of which is greater than or equal to 3 and whose sum is a specified even integer greater than 6.

[0205] How it is Played:

[0206] A complete pack of 54 cards is used. (Here 2 or the Uwa strange card is excluded). The dealer deals 6 cards to each of the players one at a time. The rest of the pack is placed face down to form the general pool. Before the cards are dealt out by the dealer, the players agree on the number which the two cards must add up to. This number is to be called the Target Sum.

[0207] The Target Sum could be arrived at by

[0208] each of the players taking turns to suggest an Even number which is adopted if supported by at least ⅔ of the players. At the beginning of the next game, another player takes up the task of suggesting the Target Sum until everyone has made an agreeable suggestion;

[0209] the dealer turning over the initial card and the group adopting that the Target Sum is the highest even integer less than the initial card.

[0210] Proviso:

[0211] The initial card must be such that can permit a solution, that is, that can bring the game to an end. Where an inadmissible card shows up as the initial card, it is tucked inside the general pool and the next topmost card turned over. This is continued until an admissible initial card shows up. (Note that for the game to come to an end the initial card must not be less than 11).

[0212] Based on the digit root of the initial card, the first player (to the right of the dealer) is given the chance to release any unhelpful cards. A card whose face or pip value is greater than the Target Sum is automatically unhelpful and burdensome and should be done away with as soon as possible. Similarly, a card whose pip value is less than the Target sum but whose complement is non-prime is unhelpful. For example, having the card with pip value 5 is unhelpful if the Target Sum is 80, since its complement 75 is not prime. Note that in addition to having two distinct cards whose sum equals the Target Sum, a player must release all other cards in his/her hands before he/she announces that a Target Sum has been arrived at.

[0213] End of game: The game comes to an end when an announcement to the effect that the Target Sum has been arrived at is verified by other players or an umpire to be correct.

[0214] Scoring: At the end of a game, any unhelpful cards in the hands of a player are added up. The player with the highest sum of unhelpful cards scores the least mark/point while the winner (with zero unhelpful cards) scores the maximum mark/point. Players should agree on what the maximum and minimum marks should be before the commencement of the game.

[0215] A card will be deemed unhelpful, if it is not one whose face value could be added to another card's face value to arrive at the Target Sum.

[0216] 6. Respect:

[0217] Aim of the game: To earn as many cards as possible and be the first person to release all the cards in his or her hands.

[0218] Number of players: 2, 3, 4, 5 or 6.

[0219] How it is Played:

[0220] A full or complete pack of 56 cards is used. The dealer deals 6 cards to each of the players one at a time. The rest of the pack is placed face down and forms the general pool. The initial card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0221] In this game, a playable card (relative to the turned over card) is any card which has a higher social standing than the turned over card and which therefore could be released by the player. It should be mentioned that the following is the natural order of social standing with respect to this game:

[0222] Uwa is accorded the highest social standing, followed by Obodo, then Ichie, Nna, Nne, Nwata and the last or least being an ordinary card. Where such is the case, the player having a playable card and whose turn it is to play, then inherits the lower-ranked card on the active platform.

[0223] If a player whose turn it is to play does not have a playable card, he/she must release one of the cards in his/her hands and then pick the topmost card facing downwards from the general pool and pass the turn over to the next person in his/her right hand and to whom the rule just explained equally applies.

[0224] Note that in this game, Obodo can not inherit Uwa (none of the cards can), Ichie can not earn or inherit Obodo, Nna can not earn or inherit Ichie. Similarly, Nne can not inherit Nna, Nwata can not inherit Nne and an Iti can not inherit Nwata. Also, no super card can inherit a card of the same social standing. For example, an Ichie cannot inherit another Ichie, a Nwata can not inherit another Nwata etc.

[0225] End of game: The game comes to an end when the general pool becomes empty or when a player exhausts all the cards in his/her hands.

[0226] Scoring: The winner is the player who has the greatest number of cards in his/her bank

[0227] Proviso: Should the turned over card be an Obodo, it suffices, in the absence of any player having or playing the Uwa card for some player to offer or produce two Ichie cards. In the case of an Uwa being the turned over card, it will suffice for three Ichie cards from three different digit root classes to be offered or produced. Whoever does this is permitted to take all the cards on the active platform as well as the offered/redeeming Ichie cards.

[0228] Hint: This game could be used to introduce social hierarchies and chains of command in present and past traditional societies. It could also be used to interrogate and possibly challenge outmoded notions of respect, loyalty and socialisation.

[0229] 7. Immobilising the Enemy's Tanks

[0230] Aim of the game: To immobilise or destroy as many of your enemy's tanks as possible by taking their tanks captive. Here the enemy is represented by a card from a different digit root (or colour) class.

[0231] Number of players: 2, 3, 4, 5 or. 6.

[0232] How it is Played:

[0233] A complete pack of 56 cards is used. The dealer deals 6 cards to each of the players one at a time. The rest of the pack is placed face down to form the general pool. The initial card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads; followed by the player to his/her own right in that order. (On no account should the turned over card be one of the two strange cards. Where such happens accidentally, the strange card must be exchanged for another card, the strange card inserted into the general pool and the general pool properly reshuffled).

[0234] In this game, a playable card (relative to the turned over card) is any card whose digit root (or colour) is different from that of the turned over card and which is at least double the face value of the turned over card.

[0235] If a player whose turn it is to play does not have a playable card, he/she must immobilise one of his/her tanks by relinquishing one of the cards in his/her hands and then pick up the topmost card facing downwards from the general pool and pass the turn over to the next person in his/her right hand and to whom the rule just explained also applies.

[0236] Exception: An exception to the rules given here is that the Obodo or Uwa strange cards could be placed on top of any card. In a sense, this is akin to the Continental Force and the United Nations/Inter-Continental Force respectively being used to quell a state of military impasse.

[0237] Hint: That only the two strange cards in a complete pack of 56 cards could be used to end a stand-off means that the strange cards could only be called in as a last resort. Thus, other players should prevent another player from calling upon the Continental Force or the United Nations/Inter-Continental Force at his/her disposal if they feel that it is pre-mature and unnecessary.

[0238] In case of disagreement on this, whether or not these troops could be called in should be decided by a simple majority decision based on a show of hands. Should there be a tie (say 3 vs.3 votes), the person having the Uwa should have a casting/deciding vote.

[0239] Note that should it become impossible to resolve an impasse for example, if the Strange cards/forces had been prematurely utilised or squandered), decision as to who is the winner should be based on the number of tanks/forces immobilised by every player before the impasse.

[0240] 8. The Foreigners

[0241] Aim of the game: To be the first player to get rid of all the cards in ones hands.

[0242] Number of players: 2, 3, 4, 5 or 6.

[0243] How it is played: A standard pack of 54 cards is used. The dealer deals 6 cards to each of the players, one at a time. The rest of the pack is placed face down and forms the pool.

[0244] The topmost card from the pool (or any card at all from the pool) is turned over for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0245] A playable card is one which when its digit root is added to the digit root of the card on top yields either 3, 6 or 9 as the digit root of the sum.

[0246] A player who has no playable card must draw from the pool until he/she receives a playable card.

[0247] Scoring: The first player to finish his or her cards (in accordance with the rules of play) is the winner of the game. Where there are more than two players. The winner of the gets 100 points, runner-up (that is, that is the one with the next smallest numerical value gets 80, next 60 etc.).

[0248] Winning: The winner is the first player to get rid of all the cards in his/her hands.

[0249] End of the game: This occurs when a player successfully gets rid of all his/her cards (in accordance with the rules of play). Note that the last card must ‘tally’ with the uppermost card on the active platform.

[0250] Exception:

[0251] An Ichie can be placed on top of any card whether or not it coheres with the rule. When such is the case, the digit root of the Ichie becomes the operative one.

[0252] This game could be used to give practice in adding a digit root to another digit root (which illustrates that the set of prime digit root classes is not closed with respect to addition);

[0253] Variations:

[0254] A variant of this will be the same game without the above exception, or one in which any of the super cards could be the over-riding or privileged card.

[0255] Another variant of this game is one in which having a playable card entitles you to earn the pooled cards/points. Where this is the case, not having a playable card would demand surrendering one of your cards and then picking one from the pool.

[0256] 9. The Shadows

[0257] Aim: To be the first player to use every card in his/her hands and to attract the highest number of pairs of corresponding multiplicative inverses. That is to say to be the player to earn as many pair of cards as possible by recognising and correctly putting on top of the turned over card the digit root multiplicative inverse of the digit root of the turned over card. If the digit root of the product¹¹ of the face values of two cards yields the digit root 1 (that is, the identity digit root class of the prime-generating digit root classes with respect to multiplication) then the two cards are multiplicative inverses under digit root operation. For example, 31 and 79 are multiplicative inverses.

[0258] Number of players: 2, 3, 4, 5 or 6 players.

[0259] How it is Played:

[0260] A standard pack of 54 cards is used. The dealer deals a minimum of 6 cards to each of the players one at a time. The rest of the pack is placed face down and forms the general pool. The topmost card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0261] In this game, a card is playable if its digit root is the multiplicative inverse of the digit root of the turned over or topmost card.

[0262] Correctly supplying the digit root of the turned over card entitles a player to earn or take both the turned over card and its inverse. These earned cards (pairs of cards and their inverses) must be kept separate from the cards in the player's hands, that is, must be banked.

[0263] Rules and Exceptions:

[0264] There are no preferences accorded to the super cards in this game of Shadows.

[0265] A player who does not have a playable card or for some reason fails to play it, must go to the pool to pick a card.

[0266] The winner is the player who at the end of the game has the highest number of pairs of cards (multiplicative inverses) in his or bank.

[0267] End of game: The game ends when one of the players successfully exhausts all the cards in his/her hands. Specifically, the game could be used to give practice in the inverse property of the digit root operation with respect to multiplication.

[0268] This game and its variants could be used to introduce and explore some of the properties of digit root operation/arithmetic as defined herein.

[0269] 10. Two Neighbours

[0270] Aim: To be the first player to end up with at least a pair of adjacent primes.

[0271] By definition, two primes are said to be adjacent if they belong to the same digit root class and differ only by 18. Thus, for any two adjacent primes, P and Q, there does not exist another prime integer, say T, in the same digit root class which lies between P and Q. For example, the following are pairs of adjacent primes: 5 and 23; 19 and 37; 163 and 181; 113 and 131. On the other hand, 7 and 43 are not adjacent even though there does not exist any prime integer lying between the two numbers in the same digit root class 7!

[0272] Number of players: 2, 3, 4, 5 or 6.

[0273] How it is Played:

[0274] A standard pack of 54 cards is used and is first of all shuffled very thoroughly. The dealer deals a minimum of 6 cards to each of the players one at a time. The rest of the pack is placed face down and forms the general pool. The initial card is selected from the pool and is turned over by the dealer in preparation for the first person to play. The player to the dealer's right leads, followed by the player to his/her own right in that order.

[0275] A card which does not and cannot form a pair of adjacent primes must be got rid of by the player in whose hands it is found. Otherwise, it will hinder the chances of the player winning.

[0276] In this game, a card is playable if it

[0277] a) belongs to the same digit root class (that is, is of the same colour as the topmost card or the turned over card);

[0278] b) is a super card.

[0279] A player who does not have a playable card or who chooses not to play must pick a card from the pool.

[0280] End of game: The game is brought to an end when a player succeeds in having only 2-adjacent primes in his or her hands.

[0281] Scoring: A player with the greatest number of two neighbours or adjacent primes as defined herein is the winner. Any other cards/primes which cannot be paired in terms of this adjacency criterion and which are left in the hands of a player must count against the player. (A minimum of a pair is required for someone to win).

[0282] Hint:

[0283] Success in this game often depends on knowing when it is more profitable to pick a card from the pool instead of surrendering/releasing a playable card. In fact, it could be quite advantageous to collect as many cards as possible in your hands and then strategically decide which ones to release—namely, the non-adjacent primes.

[0284] 11 Name of Game: Strategic Acquisition

[0285] Number of players: 2, 3 or 4 (preferably 2 players)

[0286] Aim of the game: To acquire the greatest area (in the form of triangular shapes) out of an area which could be acquired. Usually the area is 81 square units.

[0287] Materials Needed:

[0288] a) A well shuffled pack of 56 Ichongiri's Numerical Cards

[0289] b) A sheet of graph paper with the vertical and horizontal axes properly and evenly calibrated between 0 and 9 on each of the axes;

[0290] c) Pencils or pens of different colours. The number of colours needed should be the same as the number of players or participants in the game.

[0291] Preparation and Play:

[0292] Each player should decide the colour of his/her pen before the commencement of play. The already shuffled pack should be placed face down on the active platform and next to it will be the calibrated graph paper.

[0293] 1. Each player starting with the person to the right of the person who shuffled the pack of cards picks 2 cards and uses the digit roots of the picked up cards to mark a point on the graph using his/her own pen or pencil.

[0294] 2. Where the digit roots of the cards picked up are different, it is up to the player to decide which card (and therefore digit root) should be used for the horizontal axis and which for the vertical. The chosen horizontal and vertical co-ordinates must be declared by the player before the point is plotted.

[0295] 3. The next player (to the right of the person who has just marked a point) then acts as described in 1 and 2 above. This is repeated until every player has picked three pairs of cards and therefore marked three points.

[0296] 4. Beginning with the first person who plotted a point, each of the players is given the chance to join the three points he/she has accumulated using straight lines to form a triangle.

[0297] 5. The enclosed triangular shape represents an area acquired by the player.

[0298] Rules of the Game:

[0299] a) All the 56 cards in a complete pack will be used.

[0300] b) Any used pair of cards must be put aside.

[0301] c) When all the cards on the active platform (general pool) are depleted, the already used cards must be collected and reshuffled before they are used again. Alternatively, a second pack of Ichongiri's Numerical Cards could be used (where available).

[0302] d) Any three points could be joined by corresponding lines (to form a triangular area) provided none of the lines intersects another line;

[0303] e) No line is allowed to intersect another line—not even if the two lines belong to the same competitor or acquirer.

[0304] f) A line may be permitted to pass through a point in order to be joined to another point from the same acquirer or a fellow competitor.

[0305] g) Each player must use a pen or marker which distinguishes his/her points and lines from those of the other players. (A legend could be used to index each players acquisition).

[0306] h) Uwa should be regarded as digit root 2 and Obodo as 3.

[0307] Ending the Game and Determining a Winner:

[0308] The game will come to an end when all the (64 sq. units of area) of the grid are taken up or when more than ⅞ of the whole area has been occupied;

[0309] Alternatively, players may decide (before the commencement of the game) to end the game after the first 56 cards are used up. When such is the case, the player who has acquired the greatest amount of area is the winner.

[0310] (Where an eye ball test is not capable of answering the question of who has acquired the greatest area, an approximate and mutually agreeable method of answering the question satisfactorily must be adopted, e.g. calculating the area of each of the shapes belong to a player and summing them up).

[0311] Variations:

[0312] For a variant of this game, the aim could be to acquire the greatest number of triangles within the possibility space, without regard to the area. Here the lines which make up a triangle apart from not intersecting another line must emanate from and end in an acquired point.

[0313] 12. Name of Game: Natural(s) Election

[0314] Number of players: 6 people or 6 classes or 6 parties

[0315] Aim: To earn the greatest number of votes for yourself or party.

[0316] How it is Played:

[0317] After or before a thorough shuffle of standard Ichongiri pack (of 54 cards), each player or group or players decides on which of the 6 colours he/she or they should adopt as the name of its political party. For example, assuming we have Peter, Uche, Susan, Chibuzo, Ola and Musa as players or representatives for their political parties, then Musa can decide to be represented by Blue, Ola by Brown, Peter by Red and so on.

[0318] With each person or each group's adopted colour known and noted and with all the 54 cards face down on the active platform, beginning with the person or representative to the right of the person who shuffled the cards, each person or party representative collects 6 cards (or votes) one at a time in turn; no more no less.

[0319] Next a display/counting area is marked out of the active platform, with the six colours written down in their natural order of digit root occurrence in Ichongiri's Numerical Cards and Games. Each player or party representative is then asked to place the cards in his/her hands on the active platform in accordance with the colours displayed in the counting area.

[0320] The number of cards for each of the colour groups is assumed to be the number of votes cast for or won by the individual player or representative of the party associated with that colour for that particular game. To ensure accuracy, the total number of votes cast must be added together and verified to be equal to 36. Otherwise, there should be a re-count. The winner is the person or party with the greatest number of votes.

[0321] Determining the Winner:

[0322] The winner is the colour (or party) which gets the greatest number of cards (or votes) as determined by a simple count.

[0323] Please note that it is not always that an outright winner emerges after the initial casting of votes. A likely outcome is one in which two of the players or parties got the same number of votes. When such is the case, the winner is determined by another round of voting for the two players or party representatives. For this round (which is a form of electoral college), all the 20 cards/votes which are yet to be used (including the two strange cards) are thoroughly shuffled and shared out between the two (ten each). The ultimate and undisputed winner is the person or party representative who gets the Uwa vote (that is, the vote of the whole world) as one of the votes cast for him/her or his/her party.

[0324] A Draw:

[0325] In the highly unlikely event of each of the players or party representatives getting 6 votes each, an impasse will be declared and another voting must take place with the same names and rules as before.

[0326] Resolving an Impasse:

[0327] The number of cards to be shared among the players or representatives with the same number of votes will vary in accordance with how many they are who tied up. For example, for two players, ten cards each (including the strangers);

[0328] For three players (6 cards each, which must include the strangers, which means 2 non-strange cards out of the unused 20 must be excluded);

[0329] For four players, 5 cards each (including the strangers);

[0330] For five players, 4 cards each (including the strangers).

[0331] 13. Uhie na Aka¹²

[0332] Number of players: 2 or 4 (but preferably 2)

[0333] Aim of the game: To earn as many of the opponent's chips as possible. The player who earns the highest number of chips is the winner.

[0334] Materials: A complete pack of Ichongiri's Numerical Cards (all 56), an 8 by 8 square grid (solid) or checkered board, 28 chips—divided equally into two colours of 14 each between the players in the case of two players or 28 chips—divided equally into 4 colours of 7 each among the players in the case of the 4-player variant of the game. The board or solid grid must be numbered from 1 to 8 along the four equal sides as shown in FIG. 10.

[0335] Description and Specification of the Board for the Board Game:

[0336] A folding square board, made from a tree, measuring 480 mm by 480 mm with 3 mm of thickness when expanded and 440 mm by 240 mm when folded. One side of the board is divided into 2 sections—one section for placing the chips and another section for storing the numerical cards whose digit roots are used as co-ordinates. The maximum play area is divided into 81 small squares measuring 40 mm by 40 m. It is also divided calibrated from 0 to 9 as shown in the diagram above. The storage area surrounds the play area and is made up of 4 rectangular strips with two of the strips measuring 60 mm by 480 mm each and the other two 60 by 360 mm. The play area is as calibrated in the sketch given above. It has an area of 129600 square mm. It is printed on green on both sides and both sides with Ultra-Violet Varnish. The digit root sign or insignia is placed around the storage area (about 6 of them).

[0337] Positioning: The players must decide whether to sit facing each other or to sit adjacent to each other in the case of two players. In the case of 4 players, two must sit on the same side opposite or adjacent.

[0338] After the usual shuffling of the cards, the whole pack of cards is placed on the active platform—somewhere outside the game board or solid grid, the player to the right of the one who shuffled the cards picks two cards from the general pool ONE AT A TIME. The digit root of the FIRST CARD stands for the horizontal co-ordinate while the digit root of the SECOND CARD should stand for the vertical co-ordinate of the position of a chip on the board. The active player will then announce HORIZONTAL: 2 and VERTICAL: 5 (where 2 and 5 are the digit roots of the first and second cards respectively, say). One of the 14 chips belonging to the active player or one of the 7 chips in the case of the 4-person variant is then brought onto the checkered board and placed at the point on the board which corresponds to the ordered pair (2, 5).

[0339] This is then followed by the next player to the immediate right of the last player who collects the next two cards from the general pool ONE AT A TIME. The digit root of the FIRST CARD stands for the horizontal co-ordinate while the digit root of the SECOND CARD stands for the vertical co-ordinate. The active player will then announce HORIZONTAL: 7 and VERTICAL: 1 (where 7 and 1 are the digit roots of the first and second cards respectively, say). One of the 14 chips belonging to the active player or one of the 7 chips in the case of the 4-person variant is then brought onto the checkered board and placed at the point on the board which corresponds to the ordered pair (7, 1).

[0340] This process of bringing out one's chips to the board—called inodu odu is a crucial part of the game and must be allowed to continue until each of the players has brought out all his/her chips. Following the inodu odu process is the task of inyocha mbara which means surveying the landscape or the board. This is a process of mentally calculating the position of the chips of the other player(s) or opponent(s) relative to the active player's own chip(s) or position(s).

[0341] Any player who has concluded the inodu odu phase of the game can then proceed to the next stage which is called ichu nta or hunting. Ideally, all the players should conclude the inodu odu phase successively—from the person who commenced the game to the last person to begin. But for some reasons which will be made clear later, this may not happen. The last person to begin may be the first to conclude the inodu odu phase and therefore the first to enter into the ichu nta phase.

[0342] Successful execution of the ichu nta phase results in the event called nnochi while non-success results in a mere nnoghari. Nnochi encapsulates the dual process of elimination and substitution. The active player whose co-ordinates (x, y) correspond to the relative co-ordinate position (x′, y′) of the dormant player eliminates the chip of the dormant player positioned at (x′, y′) and then substitutes his/her own chip in that position. On its part, nnoghari is the process of an active player moving his/her chip (based on his/her selection of two cards from the pool) from a co-ordinate position (x,y) to another co-ordinate position (x″, y″) where (x″, y″) is an empty position or a position which cannot permit the active player to implement the lethal nnochi move.

[0343] During the inodu odu phase, chips are positioned on the board using the conventional method of locating points on the first quadrant of a grid in which X and Y form orthogonal axes. However, during the ichu nta phase (during which the nnochi or nnoghari moves are executed), any movement to the right or left of a chip passes for horizontal movement or movement along the X-axis while any movement north or south of a chip will pass for movement along the Y-axis.

[0344] A player is permitted to place two chips on a spot/position—one on top of the other, if the co-ordinates of his (as announced by him or her) coincides with a spot/position on which he/she has already placed a chip. This act of doubling a chip is called ichi eze. Ichi eze simply means crowning or being crowned. It signifies the fortification of one's position. A chip which is not an eze (that is, which is not yet crowned) is an nkiti, meaning an ordinary chip.

[0345] Rules of the Game:

[0346] 1. A chip must be placed at the point of intersection of the horizontal and vertical axes of the grid; that is where the horizontal and vertical axes meet or cross each other.

[0347] 2. Strict adherence to the principles of an ordered pair in which the first card picked from the pool stands for the horizontal co-ordinate and the second for the vertical co-ordinate must apply at all times.

[0348] 3. A player must complete the inodu odu phase before he/she will be allowed to execute the nnochi move, except as in Rule 2 above.

[0349] 4. If during the inodu odu phase an active player's co-ordinates (x, y) corresponds exactly to the already taken position (x, y) of a dormant player, then the active player is free to execute the nnochi move provided that his/her announcement corresponds to (x, y) and he/she is able to spot the chance for the execution of the nnochi move. Thus, if he/she announces (y, x) instead of (x, y) and/or fails to spot the chance, then the nnochi move will not be allowed to be executed;

[0350] 5. Only a maximum of two chips can be allowed at a spot or position. Where a third co-ordinate coincides with the position of an already crowned chip (an eze), the active player must reverse the (x, y) co-ordinates into (y, x) co-ordinates. If that is not possible, then the co-ordinate will be declared unpositionable and the active player will forfeit the chance to position or execute an nnoghari move. (It must be mentioned that the phenomenon of three chips or more coinciding at a position is highly unlikely in a game which makes use of well shuffled cards);

[0351] 6. On no account will an uncrowned chip (an nkiti) be allowed to execute the nnochi move on a crowned chip (an eze).

[0352] 7. A crowned chip (an eze) is allowed to execute the nnochi move on a fellow crowned chip (an eze). When this happens, the two chips making up the eze will be moved to the previous location of the eze on which the nnochi move is executed.

[0353] 8. A crowned chip (an eze) is allowed to execute the nnochi move on an uncrowned chip (an nkiti). When this happens, it is the top chip which eliminates and replaces the displaced chip. This has the effect of uncrowning the eze.

[0354] 9. Multiple displacement is NOT permitted. That is, if an active player's co-ordinates permit him/her to execute an nnochi move on two or more distinct chips of a dormant player or players from the active player's two or more distinct positions on the board, the active player is NOT permitted to do so. Only one displacement at a time per a pair of co-ordinate point.

[0355] 10. No active player is permitted to refuse to execute an moghari move as long as his/her chip is movable. This includes an eze. Thus, no player is allowed to refuse to move a chip into another position because that might entail uncrowning his/her eze.

[0356] 11. An active player whose co-ordinate (x, y) cannot permit a valid and acceptable nnoghari move will automatically forfeit the chance to his/her chip to a new position. When such happens, the chance to play moves to the player immediate to the right of the player who has forfeited his/her chance to move a chip.

[0357] 12. At the end of a stretch called an njem, the cards must be properly shuffled before the next njem is entered into. It is the responsibility of the last active player to shuffle the cards at the end of an njem.

[0358] 13. Two cards which have been used to establish a position on the board must be placed face up on a part of the active platform outside the board. Subsequent used cards must also be placed on top of the already placed ones in the same manner.

[0359] 14. Pre-viewing what the next two cards from the pool (or a part thereof) is not permitted. When this happens the cards remaining as part of the general pool must be reshuffled by the last active player before the breach.

[0360] 15. Every move (nnochi pr nnoghari) must be executed clearly and convincingly. If a move is in doubt or is questioned by one of the players, the active player who executed the move must re-execute it.

[0361] 16. As a check against any form of tampering with the chips, the number of chips in hand (that is earned by a player) added to the number on the board for that player must equal to the number of chips allocated to that player at the commencement of the game. When there is a dispute or allegation of tampering, this basic check for fairness should be carried out. (Remember, it is only a game!)

[0362] Ending the Game and Determining a Winner:

[0363] a) The game may legitimately be brought to an end at the end of any njem, provided the number of njem is not less than 2. The maximum number of njem which must end before the game is brought to an end must be agreed before the commencement of the actual game. When the agreed number of njem is reached, the number of chips earned by each player is counted. The player who has won the greater (for 2 players) or the greatest (for 4 players) is the winner no matter the margin.

[0364] b) The game may legitimately be brought to an end the moment there are only five (chips) on the board. Here the game is continued until only five chips are left on the board without any regard to the number of njem that has taken place or which has been implemented. The player who has won the greater (for 2 players) or the greatest (for 4 players) is the winner no matter the margin.

[0365] c) The game may also be brought to an end (in the case of two players) the moment the number of seeds on the board belonging to one of the two players doubles the number of chips on the board which belongs to the other player. When this happens, the win is automatic. This is the phenomenon of mmaji abuo.

[0366] Some of the Functions¹³ of the Use of the Privileged Cards

[0367] A privileged Ichongiri Card or a combination of privileged cards—could be used to effect any of the following:

[0368] (i) bring a deadlock or an impasse to an end;

[0369] (ii) earn other cards usually preceding cards;

[0370] (iii) redeem a card or set of cards;

[0371] (iv) change the current requirement of play (called the tenor of play);

[0372] (v) override an existing rule or requirement;

[0373] (vi) impose a new rule or requirement on the game;

[0374] (vii) count more than its face value (e.g. an Ichie could count as 6 times its face value;

[0375]  an Uwa/Obodo as 18 times etc);

[0376] (viii) prevent an opponent from playing (that is, to lose his/her turn);

[0377] (ix) cause an opponent to lose some of his/her cards;

[0378] (x) take one or more cards from the general pool (as a form of burden or punishment);

[0379] (xi) bring a game to an end (and thereby serve as a winner);

[0380] (xii) entitle a player to re-enter a game (and thereby have an annulling effect);

[0381] (xiii) manifest existing traditional hierarchies and power relations which may call for an interrogation or re-appraisal of the status quo etc.

[0382] Definition of Terms Used—for both Mixed and Specific Types

[0383] Active Platform: The physical space or surface on which a player is expected to place his/her playable card.

[0384] Aka: As used in the Board Game, this means vertical length in Igbo language and corresponds to the conventional y-axis in Co-Ordinate Geometry.

[0385] Bank: The cards which a player has earned in the course of the game which is convertible to a valid currency at the end of the game.

[0386] Currency: Any object or token from which cards in a bank owned by a player could be converted into. It could be number of computers, number of megabytes of hard disk, number of cars, number of horses (notional), money (token or notional) or indeed, any thing which players agree upon as adequate for compensating a winner or distinguishing between categories of performance or success in a game.

[0387] Dealer: This is somebody (usually one of the players) who passes out or distributes cards to all the players including himself/herself (if he/she is one of the players).

[0388] Digit Root Class: This is the digit root which every card within a family (of the same colour) has as its digit root. The applicable digit roots classes are 1, 2, 4, 5, 7 and 8.

[0389] Digit Root Sign or Symbol: This is the mathematical sign

newly developed, introduced and being popularised by the inventor of the Numerical Cards and Games for families, schools and colleges—Joseph Nnamdi Ichongiri.

[0390] Digit Root: This is the result of successively adding together the digits of a number until a single number less than 10 is arrived at.

[0391] Face Value: This is the numerical value/number of a card irrespective of its digit root or digit root class.

[0392] Full or Complete Pack: This refers to the first 56 prime numbers arranged by the present inventor to be used in playing card games. It is the Standard Ichongiri pack plus the two strange cards Uwa and Obodo.

[0393] General Pool: This is the set of cards left-over after the initial cards have been dealt out to players by the dealer or what remains of this left-over at any point during a game from which a player could pick a card.

[0394] Going to the pool: To go the pool is to pick a card from the pool either because one does not have a playable card or because one considers it to be strategically more beneficial not to play at the moment.

[0395] Hand: The cards held by a player during the game and from which he/she could play.

[0396] INC: An acronym for purposes of this specification stands for Ichongiri's Numerical Cards.

[0397] Initial Card: This is the first card to be picked from the pool and turned over before the commencement of the actual play.

[0398] Medium: The object and channel—of expressing digit-roots based numerical classification This could be any of the following: card, holes, boxes, desktops, containers. Each of these could manually or electronically operated.

[0399] Pass: To miss a turn.

[0400] Play: To play a card is to take it from one's hand and use it in the game. Playing could also mean going to the pool.

[0401] Playable Card: This is a card which a player can release on to the active platform and which other members can accept as appropriate in accordance with the rules of the particular game.

[0402] Privileged Card: This is any card which within a pack or group of Ichongiri's Numerical Cards or within a particular game on which certain rights and/or privileges are conferred which are not enjoyed by other cards. Examples of privileged cards are the representative cards 1, 2, 3, 4, 5, 6, 7, 8 and 9 in the case of a Mixed Pack, the Super and Strange Cards within the Prime Pack.

[0403] Shuffle: This is a way of randomly re-arranging the pack of cards or a part thereof, before cards are dealt out to players to ensure that no player gains undue advantage over the other players.

[0404] Standard Ichongiri Pack: This is the set of the first 54 digit-root based prime numbers greater than 3 arranged by the present inventor to be used in playing card games. It consists of 4 super cards and 5 non-super cards within each of the 6 figit root classes.

[0405] Tenor of a game: The current rule which applies to a game as agreed upon at the commencement of the game or which is capable of being modified when either a strange or super card is played.

[0406] It usually is the case that playing a super card or a strange card will alter the tenor of a game.

[0407] The Constant 8 Cards: This refers to the pervasive appearance and position of the following 8 cards: Uwa, Obodo and the 6 Ichies representing each of the digit root classes. It means that it does not matter what level¹⁴ of the Ichongiri cards is in use, these 8 cards with their constant face values of 2, 3, 5, 7, 11, 13, 17 and 19 remain unchanged. Uwa Obodo Ichie Nwata Nne Nna 2 3 19 307 379 397 433 487 523 541 577 11 263 281 317 353 389 443 461 479 13 283 337 373 409 463 499 571 607 5 257 293 311 347 383 401 419 491 7 313 331 349 367 421 439 457 547 17 269 359 431 449 467 503 521 557

[0408] Uhie: As used in relation to the Board Game, this means horizontal length in Igbo language and corresponds to the conventional x-axis in Co-Ordinate Geometry.

[0409] Uwa: This is a bi-syllabic Igbo word which in English means the ‘world’. Within the scheme of Ichongiri Numerical Cards and Games, Uwa is one of the two strange cards so designated because they do not fall within the digit root classification system employed in characterising and developing the cards and the games explained in this Companion. It is symbolised by the map of the world—the globe. In a sense, it represents that of which every one of us is a part but which does not coincide with any one of us. 

1. Apparatus for playing a game including a pack of cards in which: each card has a numerical face value which is a whole number; the cards are divided into a plurality of groups such that each card in any group has the same digit root as the other cards in the same group; and the cards in each group carry common identifying means signifying that the cards belong to that group.
 2. Apparatus according to claim 1 in which the common identifying means for the cards in each group includes an indication of the common digit root of the face values of the cards in that group.
 3. Apparatus according to claim 1 or 2 in which the common identifying means for the cards in each group includes a common colour applied to each of the cards in that group.
 4. Apparatus according to any one of claims 1 to 3 in which the face value of each card of the pack is a prime number.
 5. Apparatus according to any one of the preceding claims in which, in each group, the card having a face value which is the same as the common digit root of the cards in the group carries an indication which distinguishes it from the other cards in the group.
 6. Apparatus according to any one of claims 1 to 4 in which, in each group, at least one selected card in the group is provided with an additional distinguishing feature which is common to the different groups and which is indicative of the position of the selected card in the group when the cards of the group are arranged in the order of their face values.
 7. Apparatus according to any preceding claim in which the face values of the cards are decimal numbers.
 8. Apparatus according to claim 7 in which the face values of the cards of the pack exhibit a common mathematical characteristic or function.
 9. Apparatus according to claim 8 in which the face values of the cards in the groups are even numbers.
 10. Apparatus according to claim 8 in which the face values of the cards in the groups are odd numbers.
 11. Apparatus according to any one of claims 7 to 10 in which the pack includes nine groups of cards having respective digit roots of from 1 to
 9. 12. Apparatus according to claim 8 in which the face values of the cards in the groups are prime numbers.
 13. Apparatus according to claim 8 in which the face values of the cards in the groups are square numbers.
 14. Apparatus according to any one of claims 7 to 13 in which at least selected cards of the pack carry a series of letters of an alphabet, such that the numerical position of the nth letter of the series in the alphabet is equal to the digit root of the face value of the card plus 9(n−1).
 15. Apparatus according to any preceding claim including: a playing area on which a co-ordinate system is defined by an array of reference points; and first and second sets of playing pieces which are distinguishable one from the other.
 16. Apparatus according to claim 15 in which the array of reference points are defined by the points of intersection of a network of intersecting lines marked on the playing surface.
 17. Apparatus according to claim 16 in which the network of intersecting lines is defined by a checkerboard pattern on the playing area.
 18. Apparatus according to any one of claims 15 to 17 in which the first and second sets of playing pieces have the same shape but different colours.
 19. Apparatus according to claim 18 in which the playing pieces are playing chips or counters.
 20. Apparatus for playing a game according to any preceding claim implemented as a computer programme. 